TITLE: Functor category between two small categories is not small?
QUESTION [7 upvotes]: Let $C,D$ be two small categories, i.e. small sets of objects $O_C, O_D$ and small hom-sets $\mathrm{Hom}_C(c,c')$ and $\mathrm{Hom}_D(d,d')$.
The set of objects of $D^C$ is the set of all functors $F : C \to D$, and such an object is determined by a collection of functions, namely its function $O_C \to O_D$ on objects and its functions on hom-sets $\mathrm{Hom}_C(c,c') \to \mathrm{Hom}_D(Fc, Fc')$ ; since all sets involvled here are small and the product of small sets indexed by a small set is again small, the set of objects of $D^C$ is again small. 
Now let $F,G : C \to D$ be two functors ; the hom-set $D^C(F,G)$ has for elements collections of morphisms $Fc \to Gc$, i.e. elements of the hom-sets of $D$. Thus a natural transformation is an element of the product over all $c \in O_C$ of the hom-sets between $Fc$ and $Gc$, and by the same reasoning we conclude that $D^C(F,G)$ is contained in a small set, thus small.
So why does Mac Lane tell me that the hom-set of a functor category need not be small? Or where is my mistake? Did I misinterpret something? (Page 40, right before the last paragraph.)

REPLY [9 votes]: Everything you say is correct.  I don't have Mac Lane in front of me at the moment, but I would guess he actually only assumed that $C$ and $D$ were locally small, so while each Hom-set is small, the set of objects might be large.